Theorem dveeq1-o

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 using ax-c11 . (Contributed by NM, 2-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	dveeq1-o	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( 𝑤 = 𝑧 → ∀ 𝑥 𝑤 = 𝑧 )
2		ax-5	⊢ ( 𝑦 = 𝑧 → ∀ 𝑤 𝑦 = 𝑧 )
3		equequ1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 = 𝑧 ↔ 𝑦 = 𝑧 ) )
4	1 2 3	dvelimf-o	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )